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Theorem fin23lem30 9417
Description: Lemma for fin23 9464. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem30 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem30
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fin23lem.e . 2 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
2 eqif 4283 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
32biimpi 207 . 2 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
4 simpr 477 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5 fin23lem.d . . . . . . . . . . . 12 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
65funmpt2 6107 . . . . . . . . . . 11 Fun 𝑅
7 funco 6108 . . . . . . . . . . 11 ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡𝑅))
84, 6, 7sylancl 580 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡𝑅))
9 elunirn 6701 . . . . . . . . . 10 (Fun (𝑡𝑅) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
108, 9syl 17 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
11 dmcoss 5554 . . . . . . . . . . . 12 dom (𝑡𝑅) ⊆ dom 𝑅
1211sseli 3757 . . . . . . . . . . 11 (𝑏 ∈ dom (𝑡𝑅) → 𝑏 ∈ dom 𝑅)
13 fvco 6463 . . . . . . . . . . . . . . . 16 ((Fun 𝑅𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
146, 13mpan 681 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom 𝑅 → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1514adantl 473 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1615eleq2d 2830 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅𝑏))))
17 incom 3967 . . . . . . . . . . . . . . . 16 ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏)))
18 difss 3899 . . . . . . . . . . . . . . . . . . . . . . 23 (ω ∖ 𝑃) ⊆ ω
19 ominf 8379 . . . . . . . . . . . . . . . . . . . . . . . . 25 ¬ ω ∈ Fin
20 fin23lem.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
21 ssrab2 3847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ⊆ ω
2220, 21eqsstri 3795 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑃 ⊆ ω
23 undif 4209 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
2422, 23mpbi 221 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃 ∪ (ω ∖ 𝑃)) = ω
25 unfi 8434 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
2624, 25syl5eqelr 2849 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
2726ex 401 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
2819, 27mtoi 190 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
2928ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
305fin23lem22 9402 . . . . . . . . . . . . . . . . . . . . . . 23 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
3118, 29, 30sylancr 581 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
32 f1of 6320 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω⟶(ω ∖ 𝑃))
3331, 32syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω⟶(ω ∖ 𝑃))
34 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
3533fdmd 6232 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
3634, 35eleqtrd 2846 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
3733, 36ffvelrnd 6550 . . . . . . . . . . . . . . . . . . . 20 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ (ω ∖ 𝑃))
3837eldifbd 3745 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ 𝑃)
3920eleq2i 2836 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ 𝑃 ↔ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
4038, 39sylnib 319 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
4137eldifad 3744 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ ω)
42 fveq2 6375 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑅𝑏) → (𝑡𝑣) = (𝑡‘(𝑅𝑏)))
4342sseq2d 3793 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝑅𝑏) → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4443elrab3 3521 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ ω → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4541, 44syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4640, 45mtbid 315 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)))
47 fin23lem.a . . . . . . . . . . . . . . . . . . 19 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
4847fin23lem20 9412 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑏) ∈ ω → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
4941, 48syl 17 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
50 orel1 912 . . . . . . . . . . . . . . . . 17 ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) → (( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
5146, 49, 50sylc 65 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅)
5217, 51syl5eq 2811 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅)
53 disj 4178 . . . . . . . . . . . . . . 15 (((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
5452, 53sylib 209 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
55 rsp 3076 . . . . . . . . . . . . . 14 (∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈 → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5654, 55syl 17 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5716, 56sylbid 231 . . . . . . . . . . . 12 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
5857ex 401 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom 𝑅 → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
5912, 58syl5 34 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom (𝑡𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
6059rexlimdv 3177 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
6110, 60sylbid 231 . . . . . . . 8 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) → ¬ 𝑎 ran 𝑈))
6261ralrimiv 3112 . . . . . . 7 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
63 disj 4178 . . . . . . 7 (( ran (𝑡𝑅) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
6462, 63sylibr 225 . . . . . 6 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅)
65 rneq 5519 . . . . . . . . 9 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6665unieqd 4604 . . . . . . . 8 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6766ineq1d 3975 . . . . . . 7 (𝑍 = (𝑡𝑅) → ( ran 𝑍 ran 𝑈) = ( ran (𝑡𝑅) ∩ ran 𝑈))
6867eqeq1d 2767 . . . . . 6 (𝑍 = (𝑡𝑅) → (( ran 𝑍 ran 𝑈) = ∅ ↔ ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅))
6964, 68syl5ibr 237 . . . . 5 (𝑍 = (𝑡𝑅) → ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran 𝑍 ran 𝑈) = ∅))
7069expd 404 . . . 4 (𝑍 = (𝑡𝑅) → (𝑃 ∈ Fin → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)))
7170impcom 396 . . 3 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
72 rneq 5519 . . . . . . . 8 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7372unieqd 4604 . . . . . . 7 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7473ineq1d 3975 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈))
75 rncoss 5555 . . . . . . . 8 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
7675unissi 4619 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
77 disj 4178 . . . . . . . 8 (( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ¬ 𝑎 ran 𝑈)
78 eluniab 4605 . . . . . . . . . 10 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} ↔ ∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)))
79 eleq2 2833 . . . . . . . . . . . . . 14 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈)))
80 eldifn 3895 . . . . . . . . . . . . . 14 (𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈) → ¬ 𝑎 ran 𝑈)
8179, 80syl6bi 244 . . . . . . . . . . . . 13 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8281rexlimivw 3176 . . . . . . . . . . . 12 (∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8382impcom 396 . . . . . . . . . . 11 ((𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8483exlimiv 2025 . . . . . . . . . 10 (∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8578, 84sylbi 208 . . . . . . . . 9 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} → ¬ 𝑎 ran 𝑈)
86 eqid 2765 . . . . . . . . . . 11 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
8786rnmpt 5540 . . . . . . . . . 10 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8887unieqi 4603 . . . . . . . . 9 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8985, 88eleq2s 2862 . . . . . . . 8 (𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
9077, 89mprgbir 3074 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅
91 ssdisj 4188 . . . . . . 7 (( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∧ ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅) → ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅)
9276, 90, 91mp2an 683 . . . . . 6 ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅
9374, 92syl6eq 2815 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ∅)
9493a1d 25 . . . 4 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9594adantl 473 . . 3 ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9671, 95jaoi 883 . 2 (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
971, 3, 96mp2b 10 1 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  ifcif 4243  𝒫 cpw 4315   cuni 4594   cint 4633   class class class wbr 4809  cmpt 4888  dom cdm 5277  ran crn 5278  ccom 5281  suc csuc 5910  Fun wfun 6062  wf 6064  1-1-ontowf1o 6067  cfv 6068  crio 6802  (class class class)co 6842  cmpt2 6844  ωcom 7263  seq𝜔cseqom 7746  𝑚 cmap 8060  cen 8157  Fincfn 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-seqom 7747  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016
This theorem is referenced by:  fin23lem31  9418
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